The cue ball!
The cue ball
and "8" ball are located on a pool table as illustrated. If an
inexperienced player shoots the cue and successfully sinks
the "8" in the corner pocket ,is there much danger that the
cue ball will be deflected into the other corner pocket? A
scratch is when the cue ball sinks in any pocket.
Source - Lewis-Carroll
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1 solution
ANSWER: SCRATCH The answer is: a. Every pool shark knows that when the "Q" ball hits the "8" ball, the balls move off in directions about 90 THIS degrees apart, that is, they bound off at WAY? right angles to each other. From the illustrated position of the "8" ball, the corner pockets are about 90 degrees apart, so there is great danger of scratching. But why do the balls fly off at right angles? The balls have (or should have) equal mass so their momentum is proportional to their velocity. Therefore, the vector sum of the velocity of the "Q" ball plus the "8" ball after the collision should add up to the original vector velocity of the "Q' ball before the collision. But as the sketch shows there are many ways to pick a pair of velocities which will add up to equal the original "Q" ball velocity. Which pair should you choose? Momentum is not the only thing to consider, for the balls are elastic and the sum of the kinetic energies of the balls after collision is about equal to the original kinetic energy of the "Q" ball. Now the kinetic energy of a ball is proportional to the square of its velocity and since the balls have equal mass, the square of the velocity of the "Q" ball after the collision plus the square of the velocity of the "8" ball after the collision should add up to equal the square of the original velocity of the "Q" ball before the collision. Now by the rules of vector addi- tion, we know the vector velocity of the "8" and "Q" balls form the sides of a \ parallelogram and from the Law of Momentum Conservation the diagonal of the parallelogram is equal to the original velocity of the "Q" ball. And by conservation of kinetic energy, we know the sum of 99